Rates - the school of mathematics

The ACT Math Fundamentals

Definition: A conversion factor is a ratio that expresses how many of one unit equals another unit (e.g., 1 inch = 2.54 cm).
Common Conversions on the ACT (they usually provide these in a reference table if needed, but it’s good to be familiar):
- Length: 1 foot = 12 inches, 1 yard = 3 feet, 1 mile = 5280 feet
- Weight: 1 pound = 16 ounces
- Time: 1 hour = 60 minutes, 1 minute = 60 seconds
How to use them:
- Write the given quantity as a fraction.
- Multiply by the conversion factor(s) so that unwanted units cancel.
- Example: Convert 120 seconds into minutes:
  $$$$120\times\frac{1}{60}=2$$ minutes$

Formula:

or equivalently:
$$R=\frac{D}{T},\ T=\frac{D}{R}$$
ACT Applications:
- Solving for one of the three given the other two.
- Working with different rates (walking vs. biking).
- Average speed problems (total distance ÷ total time, not average of speeds).
- Relative speed problems: when two objects move towards each other, add rates; when moving in the same direction, subtract rates.

Basic Distance Problems
- Example: A car travels at 60 mph for 2.5 hours. How far does it go?
  $$d=60\times2.5=150$$ miles
Average Speed Problems
- Example: A car travels 60 miles at 30 mph, then 60 miles at 60 mph.
  Total distance = 120 miles.
  Time=$$\frac{60}{30}+\frac{60}{60}=2+1=3$$ hours
  Average speed = $$\frac{120}{3}=40$$ mph
Relative Motion Problems
- Example: Two trains 300 miles apart move towards each other at 50 mph and 70 mph.
  Combined rate = (50+70)=120 mph.
  Time = $$\frac{300}{120}=2.5$$ hours
Unit Conversion within Rate Problems
- Example: A runner runs at 8 minutes per mile. What is her speed in mph?
  - 8 minutes=$$\frac{8}{60}=\frac{2}{15}$$ hours per mile
  - Speed = $$\frac{1}{2/15}=7.5$$ mph

Key Tips for ACT:

Always keep track of units, they guide you to the correct setup.
Be cautious with average speed problems (don’t just average the rates).
When in doubt, write down the distance-rate-time relationship and solve step by step.
Score 80% or more in all the School of Mathematics ACT Qbank Rates QUIZZES.